We give a rigorous and mathematically well defined presentation of the covariant and gauge invariant theory of scalar perturbations of a Friedmann-Lemaître-Robertson-Walker universe for fourth order gravity, where the matter is described by a perfect fluid with a barotropic equation of state. The general perturbations equations are applied to a simple background solution of $R^n$ gravity. We obtain exact solutions of the perturbations equations for scales much bigger than the Hubble radius. These solutions have a number of interesting features. In particular, we find that for all values of $n$ there is always a growing mode for the density contrast, even if the universe undergoes an accelerated expansion. Such behavior does not occur in standard general relativity, where as soon as dark energy dominates, the density contrast experiences an unrelenting decay. This peculiarity is sufficiently novel to warrant further investigation of fourth order gravity models.

We study the phase--space of FLRW models derived from Scalar--Tensor Gravity where the non--minimal coupling is $F(\phi)=\xi\phi^2$ and the effective potential is $V(\phi)=\lambda \phi^n$. Our analysis allows to unfold many feature of the cosmology of this class of theories. For example, the evolution mechanism towards states indistinguishable from GR is recovered and proved to depend critically on the form of the potential $V(\phi)$. Also, transient almost--Friedmann phases evolving towards accelerated expansion and unstable inflationary phases evolving towards stable ones are found. Some of our results are shown to hold also for the String-Dilaton action.